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SOIL & WATER MANAGEMENT & CONSERVATION

Water Erosion Estimation in Topographically Complex Landscapes: Model Description and First Verifications

^a Facultad de Agronomía e Ingeniería Forestal, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, Chile
^b Dep. of Soil Science, Univ. of Wisconsin, 1525 Observatory Dr. Madison, WI 53706
^c Cooperative Institute for Meteorological Satellite Studies, Space Science and Engineering Center, Univ. of Wisconsin, 1225 W. Dayton St., Madison, WI 53706

^* Corresponding author (cbonilla{at}uc.cl).

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
Two of the most important limitations when predicting soil movement are the natural complexity and the spatial heterogeneity of the processes. Soil erosion can vary significantly across short distances as a function of local soil properties and microtopography; but regardless of this, many erosion models assume homogeneity in topography and soil characteristics. The objective of this research was to develop a method for estimating soil loss from agricultural fields that is faithful to the complex topography and spatial heterogeneity common to managed landscapes. Sediment loss for individual storms was achieved by linking soil detachment–deposition equations adapted from the Water Erosion Prediction Project (WEPP) model to the existing water-flow subroutine in the Precision Agricultural-Landscape Modeling System (PALMS). In PALMS, sediment was routed appropriately in a two-dimensional grid, defining the pathways taken by the eroded material. Usually PALMS works on a grid-cell size of 5 to 20 m and simulates runoff and soil erosion patterns as affected by slope, soil texture, anisotropic surface roughness, soil consolidation, canopy cover, and tillage interactions with topography. In this study, PALMS and WEPP were used to simulate the sediment transport on an idealized field with a complex hillslope profile. Both models showed a consistent soil loss pattern and only minor differences in transport capacities. The models were also compared with data from actual erosion plots, where both runoff and soil loss were predicted with similar errors for both PALMS and WEPP. To illustrate the capability of PALMS, it was applied to a field with complex topography.

Abbreviations: AGNPS, Agricultural Non-Point Source • DEM, digital elevation model • PALMS, Precision Agricultural-Landscape Modeling System • RUSLE, Revised Universal Soil Loss Equation • USLE, Universal Soil Loss Equation • WEPP, Water Erosion Prediction Project

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
Many different kinds of models are used to simulate soil erosion and sediment transport. Most soil erosion models use either an index-based or a dynamic approach. Index-based models, such as the Universal Soil Loss Equation (USLE; Wischmeier and Smith, 1978) and the Revised Universal Soil Loss Equation (RUSLE; Renard et al., 1997), represent empirical fits to erosion measurements from erosion plots to accommodate the effects of climate, soil, topography, and land use. The RUSLE2 (Foster et al., 2003) is a hybrid model that uses an index-based form in which simple process equations are used to determine values for indices. Dynamic models, like the Water Erosion Prediction Project (WEPP) model (Nearing et al., 1989; Flanagan and Nearing, 1995; Flanagan et al., 2001), use equations to represent how variables describing erosion change within and between storms. These dynamic models are also referred to as simulation models because they mathematically simulate field processes through time (Toy et al., 2002).

In terms of time resolution, soil erosion models have been developed from two points of view: (i) to describe individual rainfall events, or (ii) to predict long-term effects (Merritt et al., 2003). Event-based models (e.g., WEPP) use a time step on the order of minutes to hours to describe individual rain storms. This small time step permits prediction of the size and timing of sediment discharges from hillslopes to channels. Alternatively, annual-based models like USLE, RUSLE, and RUSLE2 estimate long-term values of mean annual soil loss. Annual-based models have been used to explore multiyear trends in soil erosion as a result of changes in rainfall, vegetation, or land management (Morgan et al., 1998; Merritt et al., 2003). Additionally, many of the models that were originally applied to single events have undergone modifications to work on a continuous daily basis (Merritt et al., 2003). Examples of continuous simulation models are the Agricultural Non-Point Source (AGNPS; Young et al., 1989) or the Areal Non-Point Source Watershed Environment Response Simulation (ANSWERS; Beasley et al., 1980).

Soil erosion itself is described as a process having three stages: detachment, transport, and deposition (Foster et al., 1985; Lal, 2001). Detachment of sediment from the soil surface is caused by raindrop impact, shearing or drag force of water, or dissolution of cementing agents through chemical reactions (Lal, 2001). Detached particles are transported by flowing water and deposited when the velocity of water is decreased due to changes in slope or ground cover. Models like the USLE and RUSLE are not able to predict deposition or the pathway taken by the eroded material as it moves from hillslope sites to water bodies (Morgan et al., 1998). The RUSLE2 and WEPP include routines to estimate sediment transport and deposition, but do not consider the actual pathway of water. Since deposition depends on the particle size of eroded sediments, the WEPP and RUSLE2 incorporate sediment composition, whereas the USLE and RUSLE ignore this factor (Young and Onstad, 1976; Foster et al., 1985; Farenhorst and Bryan, 1995; Di Stefano and Ferro, 2002).

When soil is eroded, the sediment is composed of a mixture of aggregates (conglomerates of clay, silt, and sand) and primary particles (Foster et al., 1985). The importance of the aggregates was identified by Di Stefano and Ferro (2002), who noted that detachment decreases as the particle size either decreases or increases beyond the range of 20 to 200 µm. Foster et al. (1985) derived a series of relationships to predict the composition of eroded soil (aggregate plus primary particles) from knowledge of the texture of the soil matrix. The equations describe five sediment particle classes based on experimental data: primary clay, primary silt, small aggregates, primary sand, and large aggregates. Since they were published, these relationships have been used in different erosion and sediment–nutrient transport models such as Groundwater Loading Effects of Agricultural Management Systems (GLEAMS; Leonard et al., 1987), WEPP (Nearing et al., 1989), Opus (Smith, 1992), and RUSLE2 (Foster et al., 2003).

Most of the soil erosion models use the classic one-dimensional hillslope approach to represent the topography. This approach requires that hillslopes be delineated and channels identified. Each hillslope, represented as a rectangle, must have a representative length, width, and slope profile. Depending on the model, the slope can be one fixed segment (USLE and RUSLE), a series of flat-slope segments (RUSLE2), or a continuous curved slope profile surface (WEPP). Hillslopes drain into the top, left side, or right side of a channel, eventually leading to the watershed outlet (Cochrane and Flanagan, 1999). Because actual agricultural fields have complex topography, some subjectivity is inherent in assigning hillslope properties to a real field. Recently, more precise tools for soil surface spatial data collection and more powerful computers have made it practical to accommodate greater spatial and temporal resolution in erosion predictions. Therefore, a grid approach appears as an alternative to statistical methods, especially considering integration with geographic information systems (GIS), digital elevation models (DEMs), and remote sensing. Mitasova et al. (1996) demonstrated that the USLE cannot be used in a grid approach without previous identification and exclusion of areas with potential for deposition. The USLE predicted high erosion potential at the lower, concave parts of hillslopes, where deposition is usually observed. Because the influence of flow convergence was not included, the USLE predicted relatively low erosion in areas with convergent water flow and higher erosion in some areas with dispersed water flow. Botterweg et al. (2002) used the European Soil Erosion Model (EUROSEM; Morgan et al., 1998) in combination with a DEM to analyze erosion at two watersheds in Norway, demonstrating that the grid approach was capable of estimating changes in overall erosion levels relative to benefits of different agricultural management systems. Bhuyan et al. (2002) developed an application that combined the capabilities of remote sensing, GIS, and the AGNPS model to assess sediment yield from various watersheds. This study showed that the use of remote sensing along with GIS reduces the time to obtain input for the modeling process and adds to confidence in the representation of watershed conditions. Comparing estimated and measured suspended solids for major runoff events, the modeling process was effective for small watersheds (up to 145 km² [56 mi²]) with adequate available rainfall data. For larger watersheds with substantial variations of rainfall, however, this process was less satisfactory (Bhuyan et al., 2002).

Van Oost et al. (2004) described a two-dimensional model of erosion and deposition using deposition by settling and nonselective re-entrainment of deposited sediment. They emphasized deposition and thus used a simple hydrology approach (modified curve number with kinematic wave) with simple empirical soil erosion equations. Although their results were quite sensitive to curve number, with the appropriate choice they achieved excellent agreement between predicted and measured rill patterns as well as deposition location and amount. The model of van Oost et al. (2004) predicts some of the same quantities as described here, but their model is formulated in a completely different way from PALMS.

Besides all the capabilities previously mentioned, soil erosion models are most valuable when they can be used to suggest management practices that minimize soil loss and maintain crop productivity. Addressing both crop productivity and environmental protection requires comprehensive models that incorporate all the major processes that affect both activities. Creating and validating such comprehensive models is a formidable task. Molling et al. (2005) suggested that such a comprehensive model should have the following characteristics: physically based representation of important hydrological and biophysical processes, including infiltration, runoff, soil water redistribution, evapotranspiration, photosynthesis, phenology, and biomass production; continuous simulation throughout a single cropping season as well as year-round; explicit representation of topography, crops, tillage practices, and measurable heterogeneity of soils in three dimensions; incorporation of the dynamics of key time-dependent (precipitation-dependent) processes, including soil surface roughness and sealing; computational efficiency; and adaptability to various remote sensing, geospatial, and GIS data inputs.

Over the years, models that incorporate a broad suite of processes have been created, such as the Erosion Productivity Impact Calculator (EPIC; Williams, et al., 1984), WEPP, and Opus. Molling et al. (2005) considered 17 different models and the various processes that they simulate; none of the models contained all the features needed to address both crop production and its environmental consequences on agricultural landscapes. Therefore, the Precision Agricultural-Landscape Modeling System, PALMS (Molling et al., 2005), was created to simulate a wide range of processes on real agricultural fields at the scale of precision farming (5–20 m). All the major processes listed above were included in PALMS with a rigorous approach to predicting runoff from heterogeneous landscapes, but the loss of soil or chemicals was not considered. Therefore the overall objective of this research was to develop a method for estimating the distribution and amount of soil loss and sediment deposition in agricultural fields that is faithful to the complex topography and spatial heterogeneity common to managed landscapes. Although this approach may be fundamental in nature, the goal was to create a tool that is useful for management decisions that minimize soil loss and maintain or enhance production. This main objective was achieved by linking a grid-based soil-erosion prediction subroutine to the existing water-flow landscape subroutine in PALMS.

	THE PRECISION AGRICULTURAL-LANDSCAPE MODELING SYSTEM

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
The Precision Agricultural-Landscape Modeling System, PALMS, is a computer model that simulates key hydrologic and biophysical processes year-round at a level of physical realism and spatial and temporal detail sufficient to evaluate the physical and economic consequences of specific cropping, tillage, and fertilizer management strategies for productivity, profitability, and environmental consequences. The original intent of PALMS was to create a tool that could be put in the hands of the agricultural-consultant user base. The model runs on an average laptop or desktop computer and has some support software (Molling et al., 2005).

The principal reasons for creating PALMS were shortcomings in existing models related to a general lack of ponding in closed depressions, neglect of winter processes with snow and ice, and oversimplification of heterogeneous soil properties across landscapes. Clearly PALMS is built on numerous existing models to serve the purpose of a robust application model for farm-management decisions.

The Precision Agricultural-Landscape Modeling System is a combination of two models: (i) a two-dimensional, diffusive wave, runoff model (Julien et al., 1995) with ponding, and (ii) a one-dimensional, point-column, land-process or biophysical model called the Integrated Biosphere Simulator (IBIS; Foley et al., 1996; Kucharik et al., 2000), which has been extensively tested (Delire and Foley, 1999; Kucharik et al., 2000). Using a DEM, a grid with a grid-cell size of 5 to 20 m is set up on the field of interest and the one-dimensional IBIS model is run at each grid point with appropriate soil inputs using hourly weather data. Variable soil texture is incorporated in PALMS at the subfield scale and variable hydrological properties are assigned to each grid location according to values typical of its textural class. The IBIS portion of PALMS simulates balances and one-dimensional fluxes of energy, water, C, N, and biomass production. When precipitation occurs in excess of infiltration and detention storage, the diffusive-wave model is activated and rainfall is simultaneously routed over the landscape and infiltrated into the soil.

Runoff patterns, as affected by anisotropic surface roughness (caused by row tillage), tillage-angle interactions with topography, and the change of random roughness with accumulated precipitation (Zobeck and Onstad, 1987), have been simulated by PALMS. The influence of surface sealing, which has been demonstrated by various researchers (Norton et al., 1993; Assouline, 2004), is also included in PALMS. The time step associated with the runoff portion of PALMS depends on terrain slope and can be as short as 1 s. On each runoff-routing time step, the grid erosion subroutine is called in PALMS to route the sediment appropriately, predicting soil erosion and deposition over a two-dimensional grid with complex topography. We added detachment, transport, and deposition routines to PALMS and information from WEPP was used to develop PALMS erosion–deposition routines. The many differences are summarized in Table 1.

Sediment Load
Sediment load is predicted with PALMS using mainly physics-based equations. Each grid cell is divided into rill and interrill areas, and a series of nodes are distributed along the rill (Fig. 1 ). The runoff and sediment coming from the interill are delivered to the rill. Once in the rill, the sediment loads for each node are calculated, solving the detachment and deposition equations with the Euler numerical method (Gerald and Wheatley, 1989). Whether in the rill or interrill, detached soil is partitioned into the five particle size classes (k = 1–5) using the equations proposed by Foster et al. (1985). The Euler method is a simple and fast approach to predict the value of a variable based on its value at the initial condition and the rate of change at the beginning of the step (Gerald and Wheatley, 1989). Using the Euler method, PALMS calculates the sediment loads by solving the following equation:

[1]

where G_k,n+1 is the sediment load for a sediment particle size class k at node n + 1 (kg s^–1 m^–1), G_k,n is the sediment load at node n (border condition; kg s^–1 m^–1), dG_k,n/dx is the rate of change at the beginning of step n + 1, x is the distance downslope (m), and x is the node spacing or step (m).

View larger version (16K): [in this window] [in a new window]   Fig. 1. Rill and interrill partitioning and node distribution in a PALMS grid cell.

The rate of change at the beginning of the step (dG_k,n/dx) is provided by the same steady-state sediment continuity equation used in WEPP (Nearing et al., 1989):

[2]

where D_r,k,n is the rill erosion rate (kg s^–1 m^–2) for a sediment particle size class k at node n, and D_i,k,n is the interrill sediment delivery to the rill (kg s^–1 m^–2). Both D_r,k,n and D_i,k,n are computed on a per-rill-area basis so that G_k,n is solved on a per-rill-width basis. The value of D_i,k,n is independent of x within a grid cell, and always positive or zero, whereas D_r,k,n is positive for detachment and negative for deposition, resulting in two sets of equations for describing rill processes.

The WEPP model solves Eq. [2] in the detachment mode using numerical integration down the hillslope and uses an analytical solution if deposition occurs. This approach used in WEPP works for hillslopes but will not work in a grid network. The Euler method is used in PALMS to do numerical integration for detachment, deposition, and transport. Substituting the sediment continuity equation, Eq. [2], in the Euler solution, Eq. [1], PALMS calculates the sediment load G_k,n+1 as follows:

[3]

The rill and interrill erosion rates are calculated based on the equations developed for WEPP. The interrill erosion rate (D_i,k,n) for a sediment particle size class k at any node n is calculated as follows (Foster et al., 1995):

[4]

where k_i is the interrill erodibility factor (kg s m^–4), I_e is the effective rainfall intensity (m s^–1), _i,n is the interrill runoff rate (m s^–1) at node n, and SDR_k,n is the sediment delivery ratio for a particle size class k and node n, which is computed as a function of the random roughness of the soil surface, the fall velocity of the individual particle size class of sediment, and the particle size distribution of the sediment. The parameter F_nozzle is the adjustment factor to account for sprinkler irrigation nozzle impact energy variation, R_s is the spacing of the rills (m), and w_n is the rill width at node n (m).

The sediment deposition in the rill (D_r,k,n) for a sediment particle size class k, at any node n, is calculated as follows:

[5]

where ß is a raindrop-induced turbulence coefficient (dimensionless), V_f,e is the effective fall velocity for the sediment (m s^–1), V_f,k is the fall velocity for a sediment particle size class k (m s^–1), T_t,n is the total sediment transport capacity at node n (kg s^–1 m^–1), T_k,n is the sediment transport capacity for the sediment particle size class k at node n (kg s^–1 m^–1), G_t,n is the total sediment load at node n (kg s^–1 m^–1), and q_n is the flow discharge per unit rill width at node n (m² s^–1). The last term on the right in Eq. [5] is a partitioning function used to divide the total sediment deposition among the five particle size classes.

Following the same procedure described for the WEPP model, the fall velocity is computed in PALMS for each sediment particle size class k as follows:

[6]

where d_k and SG_k are the particle diameter (m) and specific gravity (unitless) for a particle size class k, g is the acceleration of gravity (m s^–2), and is the kinematic viscosity of water (m² s^–1). Just as in WEPP, the equations derived by Foster et al. (1985) are used to compute diameter and specific gravity for each of the five particle size classes as a function of primary clay, silt, and sand fractions and organic matter content of the surface soil horizon (Nearing et al., 1989).

The effective fall velocity, V_f,e, is computed in PALMS by using Eq. [6] with an effective diameter and effective specific gravity of the sediment at the point of detachment. Adapting the relationship initially developed for WEPP, the effective diameter is computed from the five particle size classes using the equation in Foster et al. (1995, Eq. 11.3.12). In the original equation, the WEPP model used only the three smaller particle size classes to calculate V_f,e (Foster et al., 1995). Using only three particle size classes in WEPP was a methodology "still under evaluation and future refinements of the WEPP technology may include changes to this lumped function or implementation of a different procedure which uses all particle size classes" (Foster et al., 1995).

The soil detachment in the rill, D_r,k,n, as a function of particle size class k and node n is calculated as follows (Foster et al., 1995):

[7]

[8]

where D_c,n is the detachment capacity by rill flow at node n (kg s^–1 m^–2), k_r is the rill erodibility factor (s m^–1), _r,n is the flow shear stress acting on the soil particles at node n (kg m^–1 s^–2), and _c is the rill critical shear stress (kg m^–1 s^–2).

Substituting Eq. [5] and [7] in Eq. [3], for deposition G_k,n+1 is calculated as

[9]

and for detachment as

[10]

Whether for detachment or deposition, Eq. [9] and [10] provide five independent equations for the five particle size classes. The net detachment or deposition is calculated from the difference in sediment loads between two consecutive nodes as follows:

[11]

where E_k,n+1 is the net detachment or deposition for a sediment particle size class k at node n + 1 (kg m^–2), and t is the routing time step (s). At each node, the decision for using the detachment or deposition equation is made based on the ratio between the total sediment load and the total transport capacity in the previous node. The detachment equation is used to calculate G_k,n+1 when G_t,n < T_t,n and _r,n+1 > _c; otherwise, the deposition equation is used if G_t,n > T_t,n.

The numerical solution for each particle size with PALMS is not exactly the same as the particle size estimation used in WEPP, but PALMS represents an attempt to create a numerical equivalent to WEPP that is applicable on a two-dimensional landscape. For hillslopes, PALMS and WEPP showed similar predictions.

Sediment Transport Capacity
The equation for sediment transport capacity is a very fundamental part of an erosion model (Foster, 1982). Whether or not deposition is calculated depends directly on the magnitude of the transport capacity relative to the sediment load made available by the detachment process. The transport capacity at each node in the rill is calculated in PALMS using a modification of the Yalin (1963) equation. The original Yalin equation, designed for sediment with uniform particle-size distribution (a single particle size), is described by Finkner et al. (1989). The Yalin equation for nonuniform sediment, which was described by Foster (1982), is used in PALMS to compute the transport capacity at each grid-cell node with two modifications proposed in WEPP (Foster et al., 1995). The modifications were based on extensive testing of the WEPP model for a large range of different soil types and measured field erosion data (Foster et al., 1995). To apply the Yalin equation as described by Foster (1982), the sediment load for each of the particle size classes is summed to obtain the total sediment load. The first modification consists of the calculation of a weighted average of the sediment for each particle size class, where the weighting function is the mass fraction of sediment within each class. The second modification consists of an empirical adjustment function for soils with sand content >50%. The total (T_t,n) and individual (T_k,n) transport capacities used in Eq. [5] are multiplied by an adjustment factor calculated as follows (Foster et al., 1995):

[12]

where "sand" is the fraction of sand in the surface soil, and t_adj is the empirical adjustment factor, which is limited to a minimum value of 0.30.

Among nine different transport formulae, the Yalin equation has proved to be the best transport capacity predictor for shallow flows (Alonso et al., 1981), and PALMS uses it to calculate the value at each node. The WEPP model used a slightly different approach with a simplified relationship to predict the transport capacity (Nearing et al., 1989). The WEPP uses the Yalin equation and its modifications only to compute the transport capacity at the end of the slope (Nearing et al., 1989). Following the method outlined by Finkner et al. (1989), this transport capacity is used to calibrate a transport coefficient, which is used with the local shear stress to predict the transport capacity to the top of the hillslope (Ferro, 1998; Flanagan and Nearing, 2000). Nevertheless, the differences between the simplified relationship used in WEPP and the Yalin equation as used in PALMS are minimal (Finkner et al., 1989), with the direct use of the Yalin equation being preferred.

Soil Consolidation
Diverse studies have demonstrated the agricultural, hydrological, and environmental importance of soil consolidation (Norton et al., 1993; Assouline, 2004). Soil consolidation is a complex phenomenon not only related to the soil–rainfall system, but also to physical, chemical, and biological factors affecting the soil's resistance to destruction (Assouline, 2004). Models like WEPP represent soil consolidation by modifying the bulk density and the sealing and crusting in the soil surface. The bulk density influences water retention and infiltration at the soil surface, and the sealing and crusting reduce the susceptibility to water erosion (Alberts et al., 1995). In PALMS, the effect of soil consolidation on soil erosion is simulated using a modified version of the WEPP equations to reduce the susceptibility to water erosion by changing the critical shear stress and the rill and interrill erodibility factors.

The critical shear stress, _c, is a parameter used in the rill detachment equation, and is the shear stress below which no soil detachment occurs (Alberts et al., 1995). Based on WEPP, in PALMS _c is calculated as follows (Alberts et al., 1995):

[13]

where _cb is the baseline critical shear stress (kg m^–1 s^–2), and c_s,rr, c_s,sc, and c_s,ft are the adjustment factors for surface roughness, soil consolidation, and freeze–thaw effects, respectively. The baseline critical shear stress and the surface roughness and freeze–thaw adjustment factors are calculated with the same procedure as in WEPP (Alberts et al., 1995). The soil consolidation factor, c_s,sc, is calculated as a function of time alone in WEPP. This treatment of soil consolidation as a function of time only is a simplification that may be appropriate only under "average" climatic conditions. The treatment of soil consolidation in PALMS is described below.

The rill erodibility factor, k_r, is a measure of soil susceptibility to detachment by concentrated flow, and is often defined as the increase in soil detachment per unit increase in shear stress of clear water flow. Interrill erodibility, k_i, is a measure of sediment delivery rate to rills as a function of rainfall intensity and runoff rate (Alberts et al., 1995). Just as in WEPP, base values for k_r and k_i are adjusted for factors that influence the resistance of the soil to detachment:

[14]

[15]

where k_rb is the baseline rill erodibility (s m^–1) and k_ib is the baseline interrill erodibility (kg s m^–4). The rill multiplicative adjustment factors c_r,res, c_r,dr, c_r,lr, c_r,sc, and c_r,ft account for incorporated residue, dead roots, live roots, soil consolidation, and freeze–thaw cycles, respectively. The interrill multiplicative adjustment factors c_i,can, c_i,gc, c_i,dr, c_i,lr, c_i,sc, c_i,sl, and c_i,ft are for canopy effects, ground cover, dead and live root biomass, soil consolidation, interrill slope adjustment, and freeze–thaw cycles, respectively.

The baseline rill and interrill erodibilities are calculated just as in WEPP (Alberts et al., 1995). With the exception of the coefficients related to soil consolidation (c_r,sc and c_i,sc), all the multiplicative adjustment factors are calculated just as in WEPP. The soil consolidation factors are calculated in PALMS with the following equations:

[16]

[17]

where b is a decay function. Just as in WEPP, the coefficients a_r and a_i are calculated as follows:

[18]

[19]

where k_r,cons and k_i,cons are the consolidated soil erodibility for the rill and interrill, respectively. These consolidated erodibilities are calculated as described in WEPP (Alberts et al., 1995).

In WEPP, the decay function b is calculated based on the number of days since last soil disturbance. For applications of PALMS to realistic landscapes, b is calculated using a function that is consistent with the calculation of the effect of crusting on infiltration. For landscape applications, PALMS calculates b based on the relation between soil clay content, accumulated rainfall energy, and the formation of restricted crust layers at the surface of cultivated soils proposed by Smith (1992). Using Smith's approach, b is calculated in PALMS as follows:

[20]

where c and d are parameters used to represent the decay function. According to Smith (1992), the reduction in the soil surface hydraulic conductivity due to crust formation is assumed an exhaustion function related to the accumulated amount of rainfall energy intensity, and can be predicted by solving the following equations for c and d:

[21]

[22]

where c is the change in the soil surface hydraulic conductivity due to crust formation and EI_accum is the accumulated amount of rainfall energy intensity (MJ cm ha^–1 h^–1) as described for the USLE factor (Wischmeier and Smith, 1978). The term d is a function that represents the effect of the clay content on the ultimate crust hydraulic conductivity that can be formed by the action of rainfall.

Using Smith's equations, PALMS considers the effect of the clay content and accumulated rainfall energy when calculating the soil consolidation factors; however, the revised equations in PALMS predict ultimate values for c_r,sc and c_i,sc, for large accumulated rainfall energy, that are the same as calculated in WEPP for long times since soil disturbance. Either because of accumulated rainfall energy or the number of days since the last soil surface disturbance, the ultimate values for c_r,sc and c_i,sc are a_r and a_i, respectively, calculated with either PALMS or WEPP. As an example, Fig. 2 shows the changes in the soil consolidation factor in PALMS as a function of the accumulated amount of rainfall energy intensity. Three different soil textures are included for comparison for the interrill estimations.

View larger version (26K): [in this window] [in a new window]   Fig. 2. Change in the rill and interrill soil consolidation factor as a function of the accumulated amount of rainfall energy intensity (EI). The interrill factor responds to soil texture, so PALMS estimations are compared for three different soils.

Roughness Coefficients for Cropland Rills
Using the same approach developed for the WEPP model, in PALMS the shear stress in rills is partitioned into two parts: one part that acts on the soil to cause detachment, and another portion that acts on exposed residue or other surface cover and thus is not active in terms of soil detachment. The portion of the shear stress that acts on the soil and causes erosion is proportional to the ratio of the friction coefficient for the soil to the total friction coefficient (soil plus cover). If cover exists in the rill, the portion of total shear that acts on the soil is only a fraction of the total shear stress in the rill. The total friction coefficient for rill areas, f, is given as (Gilley and Weltz, 1995):

[23]

where f_sr is the friction coefficient for rill surface roughness, f_cr is the friction coefficient for rill surface residue, and f_live is the friction coefficient for living plants, which act to slow runoff. In the WEPP model, f_sr is assumed equal to 1.11, based on experimental data from 11 sites located throughout the eastern USA (Gilley et al., 1990). In PALMS, f_sr is assumed related to the soil texture in the soil surface, and is calculated using the following equation:

[24]

where "clay" and "sand" are the clay and sand fractions in the soil surface, respectively. This equation is taken from the source code of the WEPP model (Version 2004.701) and based on experimental data where soil texture was found to significantly influence rill friction factors, so a regression equation was developed to relate hydraulic roughness to sand and clay fractions (J. Gilley, USDA-ARS, Lincoln, NE, personal communication, 2006).

Rill and Interrill Processes
Sediment in the runoff water is generated in rill and interrill areas when there is runoff on a grid cell. Sediment from interrill areas is delivered to the rill, where it can either stay suspended in the runoff, or be deposited. Sediment in the rill does not move to interrill areas. In PALMS, flow between grid cells is assumed to occur only by rills.

The rill density and width are based on experimental studies conducted by Gilley et al. (1990), who suggested use of a rill density of about 1.0 rill m^–1. The rill flow rate, which is assumed to be the same in each rill, was calculated based on the rainfall excess and rill density. Rill width is estimated using the following regression equation developed by Gilley et al. (1990):

[25]

where w_n is the rill width at node n (m), and Q_r,n is the rill flow rate at the same node (m³ s^–1). The effects on rill geometry of soil layers with varying density were not considered in the model.

In PALMS, the widest rill width, which is created by the largest previous event, is preserved so that subsequent events use this rill width until a larger event occurs and establishes a new width to be "remembered" by the model. Of course, tillage events erase all rills so the process of establishing the widest rill begins after each tillage event. For fields where broad sheet flow can occur, such as closely seeded grasses or sod, the rill width can be set equal to rill spacing, resulting in the lowest erosive condition.

With PALMS, the same equations are used to describe sediment transport in rills and larger concentrated-flow zones; these rill equations include the effect of nonerodible layers. Usually the depth of nonerodible layers is deeper than the rills cut, so rills often are not affected by nonerodible layers; however, the depth of concentrated-flow zones often is limited by such nonerodible layers. Although channel equations in WEPP are slightly different than rill equations in the way deposition is handled, using the rill equations gives a reasonable approximation to the effects of concentrated-flow zones over complex landscapes.

Water and Sediment Transport across the Landscape
Water Movement
Surface water in PALMS is split into three categories: (i) detention storage, (ii) pond, and (iii) overland flow. A certain amount of water can be stored on the surface of any grid cell. The depth of this water (assumed to be uniformly distributed) depends on the average detention storage, which in turn is a function of tillage type, tillage angle with respect to slope, and slope gradient. Because surface roughness decreases with accumulated precipitation, detention storage decreases, too.

If more water is delivered to a grid cell (in rain or run-on) than can evaporate and infiltrate, the detention storage will begin to fill. When the detention storage capacity is reached, the runoff routing algorithm is invoked. Runoff is routed on a smaller time step than the rest of the model processes. This feature allows water to move across the landscape at a physically realistic rate. Routing time steps are variable depending on slope and water depth, but are on the order of seconds. Water is routed according to the diffusive wave equation, with an additional momentum term for deep water and a special numerical algorithm to stabilize the matrix of flow rates (Molling et al., 2005). The rate at which water moves across the surface is a function of surface roughness, slope of the water surface, and water depth.

Water from a single grid cell may move to or from one or more of the cell's four adjacent neighbors. Water moving onto the cell of interest from a neighboring cell is called run-on, while water moving from the cell of interest to a neighboring cell is called runoff. Both run-on and runoff can occur on a single grid cell during one routing time step. Runoff minus run-on (net runoff) is accumulated with each routing time step and summed over many time steps to get the net runoff across longer time intervals.

In some landscape configurations, an enclosed portion of the topography will allow water to accumulate, but not run off. We refer to this as ponding. The diffusive wave equations we use have been modified slightly to allow numerical stability in ponding situations. A true pond (water that does not move out of the model domain horizontally) can form in any depression in the topography. The presence of a pond can effectively change the surface topography (elevation of terrain plus water) so that areas of the domain contributing runoff to other areas can change dynamically. If a depression high on the landscape is not filled with water, upslope areas will not contribute runoff to the bottom of the slope. Water will accumulate in the depression and stay there. When the depression fills, however, water flowing from upslope areas can flow across the depression through the pond and continue down the rest of the slope.

The runoff routing approach used in PALMS is based on the work of Julien et al. (1995), but their solution produced too much numerical noise (checkerboard patterns) to use with sediment transport, because deposition is extremely sensitive to slight anomalies in the flow. Molling et al. (2005) describe a method to stabilize the numerical solution for overland flow. Without this stabilizing numerical algorithm, simulating deposition would be impossible.

Sediment Transport
Sediment load is a function of soil type, water velocity, surface and plant-stem roughness, and a few other factors. In general, if the sediment load in run-on is less than the transport capacity for the amount of runoff, there will be detachment in the cell. If sediment load in the run-on is more than transport capacity for the amount of runoff, there will be deposition in the cell. As in WEPP, either detachment or deposition can occur at individual nodes in a grid cell, but not both together at a node; this means that occasionally both erosion and deposition can occur at different nodes in a grid cell, but individual grid cells have either net erosion or net deposition. Because the soil type, water velocity, and roughness differ in various parts of the landscape being simulated in PALMS, detachment and deposition will vary across the landscape, both across the slope as well as down the slope.

Since run-on can occur from more than one cell, sediment load for the run-on is calculated as a mix of the sediment loads of the contributing grid cells. For example, if the north cell contributes water with one unit of sediment in 10 units of water (0.1 sediment load), and the east cell contributes water containing three units of sediment in 12 units of water (0.25 sediment load), then the total run-on contains four units of sediment in 22 units of water (0.18 sediment load). Loads are computed for each sediment size class separately. Runoff to neighboring cells has the same sediment load, even though the runoff volume may differ among the neighbors.

As sediment is moved along the landscape with the runoff, detachment or deposition can change the elevation of the soil surface and the surface soil texture. While this is observed in the field, the current version of PALMS does not change surface elevation or surface texture in response to the soil erosion process. This feature may be added in future versions.

Parameter Values for PALMS
The grid erosion subroutine of PALMS draws heavily on equations from the WEPP model, the main difference being extension to two dimensions for simulating realistic landscapes. Process-level models, such as WEPP and PALMS, have many parameters. In this study, no parameters were adjusted to improve the agreement between model predictions and measurements. All parameterizations have been taken from the literature. The sources of all the equations described in the derivation above are in the List of Symbols below.

	RESULTS

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
The predictions of soil loss from PALMS were compared with predictions from WEPP under conditions when hillslope approximations are known to be valid; this comparison provides confidence in the faithful use of WEPP parameterizations in PALMS. The first test consisted of simulating an idealized field exposed to detachment, transport, and deposition. The same field was simulated with PALMS (Version 2.9.9_20060531) and WEPP (Version 2004.701), and both models were compared in terms of the sediment transport capacity and soil-loss predictions for the entire slope length. Comparison also included the particle size distribution in the sediment exiting the field. The second test consisted of a comparison with actual rainfall–runoff–erosion events recorded in erosion plots. The data set included 11 different soils, with a wide range of soil textures and slope gradients. Both models were used to simulate the erosion rates using the actual soil texture, slope gradient, and rainfall intensity as inputs.

Idealized Field
The transport of soil was simulated by PALMS and WEPP on an idealized field with a complex hillslope profile. This idealized field was designed to include detachment, transport, and deposition of soil along a hillslope appropriate for the application of WEPP. A comparable hillslope was created using grid squares appropriate for PALMS. To have equivalent runoff rates in both models, infiltration was zero in both models.

The idealized field used in this simulation consists of a hillslope with three different slope segments (Fig. 3 ). The first 100-m-long (10% slope) segment is connected to a second segment 70 m long (4% slope), which in turn is connected to a third segment 30 m long (1% slope). This configuration was selected to promote detachment in the first half of the hillslope and provide the conditions for deposition in the second and third segments. The same profile was used in PALMS and WEPP and transitions between slope segments were smoothed according to the algorithm in WEPP. A uniform silt loam soil (15% clay, 20% sand) was used for the entire hillslope. In both models the same runoff event was simulated, with a runoff rate of 20 mm h^–1 for 1 h. The grid-cell size used in PALMS was 5 m and the number of nodes per cell was 50.

View larger version (44K): [in this window] [in a new window]   Fig. 3. Slope (S) profile used for the idealized field.

View larger version (19K): [in this window] [in a new window]   Fig. 4. A comparison of the sediment transport capacities calculated with PALMS and WEPP in an idealized field with a silt loam soil having 30 d since the last soil disturbance.

View larger version (18K): [in this window] [in a new window]   Fig. 5. A comparison of PALMS and WEPP detachment and deposition rates predicted in an idealized field with a silt loam soil having 30 d since the last soil disturbance. Positive values of soil loss indicate detachment and negative values indicate deposition.

The output from PALMS and WEPP were also compared for numerous other quantities: sediment interrill contribution, average of the soil loss in net detachment areas, maximum soil loss, distance downslope to maximum soil loss, total sediment load, sediment leaving the profile, and the fractions of sediment in various size classes in sediment exiting the hillslope (Table 2, Fig. 6 ). Although PALMS and WEPP use the same equations to calculate the sediment interrill contribution, the slope factor used in the equations is calculated in a different way. In PALMS, the sediment interrill contribution is calculated for each grid-cell node based on the local slope; WEPP calculates only one sediment interrill contribution for the entire profile, using an average slope. This explains the differences in the interrill values presented in Table 2. Sediment interrill contribution also affects the detachment rate in the rill. Detachment rate has an inverse relationship with the sediment load, which is affected by the sediment mass delivered from the interrill area. The larger interrill contribution calculated in PALMS reduces the breach between the transport capacity and sediment load, leading to a reduction in the detachment rate in the rill. Since PALMS calculates the shear stress based on the actual rill width at each node, however, PALMS ended increasing the average of the soil loss in the net detachment areas. Even with these differences, both models agree in the maximum soil loss (10 kg m^–2) and the location where this maximum is predicted (90 m downhill).

View larger version (28K): [in this window] [in a new window]   Fig. 6. Comparison of PALMS and WEPP predicted particle-size distribution in flow exiting an idealized field.

Erosion Plots
Soil-loss predictions for PALMS and WEPP were tested on a set of rainfall simulation data from Gilley et al. (1990). The purpose of this comparison with data from actual erosion plots was to test how well the models could compute variations in sediment load due to variations in slope gradient and soil texture. These data from Gilley et al. (1990) were used to assist with parameterization of WEPP.

Gilley et al. (1990) studied the rill density and rill flow rates during rainfall simulation tests conducted at 11 sites located throughout the eastern USA. These soils were selected to cover a broad range of physical, chemical, biological, and mineralogical properties. Each soil was considered to be of regional or national importance. The study areas were located on uniform slopes having homogeneous soil characteristics. Either corn (Zea mays L.) or a small grain was planted the previous year. All surface residue was first removed, the area was moldboard plowed 3 to 12 mo before the test was conducted, and then disked and raked by hand immediately preceding testing. Two plots, 3.7 m across the slope by 10.7 m long, were established at each site using sheet-metal borders. A portable rainfall simulator was used to apply rainfall at an intensity of approximately 57 mm h^–1 for 1 h. A trough extending across the bottom of each plot gathered runoff, which was measured using a flume with a stage recorder. Runoff samples for sediment content determinations were collected at 5-min intervals and the runoff and soil loss were calculated. Among the 11 sites, the slope gradient ranged from 3.8 to 9.8%. The clay content in the soil surface ranged from 3 to 39%. The sand fraction ranged from 2 to 86%. Five soil textures were included in this data set: clay loam, silt loam, loam, sandy clay loam, and loamy sand.

For the first comparison, no calibration was performed. Neither the rill and interrill erodibility factors nor the critical shear stress were adjusted. Figures 7 and 8 show measured vs. predicted runoff and soil loss using PALMS and WEPP. Both model estimations show a similar distribution for runoff. The RMSE, R², and model efficiency (ME; Nash and Sutcliffe, 1970) were computed for the estimations (Table 3). The ME can range from – to 1, and the closer the value is to 1, the better the individual predictions. A negative ME value results when the difference between observed and predicted values is greater than between observed and the mean of observed values. Both runoff and soil loss were predicted with similar errors using either PALMS or WEPP, but the R² and ME are slightly higher for PALMS for runoff or soil loss.

View larger version (21K): [in this window] [in a new window]   Fig. 7. Measured vs. PALMS and WEPP predicted runoff for data taken from Table 2 of Gilley et al. (1990).

View larger version (24K): [in this window] [in a new window]   Fig. 8. Measured vs. PALMS and WEPP predicted soil loss for data taken from Table 2 of Gilley et al. (1990).

View larger version (23K): [in this window] [in a new window]   Fig. 9. Measured vs. PALMS and WEPP predicted soil loss for data taken from Table 2 of Gilley et al. (1990) after runoff calibration.

View larger version (57K): [in this window] [in a new window]   Fig. 10. Output from a PALMS model run simulating a 28.2-mm rainfall runoff event on a 3.7-ha field near Saukville, WI (grid cells shown are 5 by 5 m; topography is exaggerated 14x; north is to the left in the figure). (a) Net erosion for the entire runoff event. Darker shades are net erosion; lighter shades are net deposition. Values range from –4.28 kg cell–1 (deposition) to 0.25 kg cell–1 (erosion). The position of the runoff and sediment collector is shown with an asterisk. (b) Number of rills per cell formed by this runoff event.

Figure 10a shows the event total net erosion per grid cell (5- by 5-m size) as calculated by PALMS. Dark areas indicate net erosion, while light areas indicate net deposition. Values range from the extremes of –4.28 kg cell^–1 (which is equivalent to 1.7 Mg ha^–1 deposition) to 0.28 kg cell^–1 (0.1 Mg ha^–1 erosion), with a mean of 0.03 kg cell^–1 (0.01 Mg ha^–1). The erosion map produced by PALMS depicts net erosion occurring in convex areas, and deposition occurring in concave areas, as one might expect from the topography. Observed runoff at the collector was 4.23 m³; PALMS simulated 4.34 m³. Event-averaged sediment concentration measured in the collector was 0.5 g L^–1; PALMS simulated 0.7 g L^–1 using the usual equations for runoff that do not include momentum equations. When momentum equations are also used to estimate runoff, the sediment concentration in the collector was predicted to be 0.6 g L^–1. These sediment concentrations correspond to an average erosion in the collector area of 0.02 Mg ha^–1 observed, and 0.03 Mg ha^–1 simulated without momentum equations or 0.025 Mg ha^–1 with them. Because this is a relatively small event of short duration and modest intensity, most of the erosion is interrill erosion so the erosion is relatively constant down the flat slope until deposition occurs in the rills. With larger events, such as 88 mm of rain with a maximum intensity of 108 mm h^–1 on an 18% slope, erosion increases by a factor of two to three or more across a downslope distance of 50 m without convergence of slopes.

Figure 10b shows the number of rills per cell as simulated by PALMS. As water flows down the slope and converges, PALMS simulates the more numerous, narrow rills combining to form fewer, wider rills. When the ratio of rill width to rill spacing exceeds 0.1, the number of rills per grid cell is reduced by one rill. Thus as flow on a grid cell increases, the number of rills on that grid cell decreases. The cells in which the rills eroded down to the nonerodible layer (0.2 m in this simulation) are similar to the set of the cells in the figure that contain only one rill. This defines a concentrated-flow zone.

Sediment yield was calculated for this event using PALMS with a fixed rill spacing of 1 m and a fixed spacing of 5 m across the entire field; the results were within 10% of each other and the results from the variable-rill-spacing example shown in Fig. 10. Also, sediment yield was about 10% less when the widest rill width was preserved in PALMS vs. instantaneous rill width calculation to always maintain full rills. Clearly for this single event, the detail characteristics associated with rills have a minor effect. This may occur because rill erosion is relatively small on this well-consolidated soil so that deposition of soil eroded from the interrill is the dominant rill process. The effects of preserving rill width from the largest event might be greater during a season with multiple rainfall events or on fields with different topographic features.

The challenge associated with using a hillslope model is apparent from Fig. 10. How many hillslope profiles with how many slope segments of what steepness and length are required to "feed" how many channels of what length to simulate this field? With PALMS, such subjectivity is largely removed.

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
The PALMS was implemented with a soil-erosion subroutine to predict detachment, transport, and deposition in topographically complex landscapes. The soil-erosion subroutine was built based on the existing WEPP equations for soil erosion, but many differences between WEPP and PALMS exist.

The comparison of soil losses calculated with PALMS and WEPP show only minor differences on an idealized field. Discrepancies are negligible considering the differences between the two models. Likewise, the tests with actual data from soil plots demonstrate that, on uniform soil plots, PALMS performs similarly to WEPP when computing soil loss as a result of variations in runoff rates, slope gradient, and soil texture.

Given topographic and soil maps, runoff and soil loss can be simulated routinely with PALMS on realistic landscapes, including concentrated flow zones in convergence areas. Creating equivalent simplified landscapes to simulate soil loss, or piecing hillslope and concentrated-flow models together is no longer necessary. Since PALMS handles heterogeneous soils, the spatial distribution of infiltration associated with variable texture is accommodated as well as the effects of the spatial distribution of texture on runoff and erosion processes including ponding. Although such variability might be possible to handle in other erosion models in principle, such an effort is too time consuming to be done except as a demonstration. With PALMS, such heterogeneity is routinely accommodated with little effort once the soil and topographic maps are available.

The interaction between tillage direction and topography is included in PALMS so differences in soil losses with various tillage orientation choices on the actual landscape are accommodated. For example, the difference in erosion or sediment yield associated with north–south tillage vs. east–west tillage can be simulated for particular fields and compared with contour tillage, which is considerably more difficult for farmers to execute.

Soil consolidation factors as a function of cumulative rain or time are among the most important factors affecting erosion. The time function for soil consolidation used in WEPP is a reasonable average relation for soil consolidation based on climatology for some particular area, but it cannot be correct in general. If intense rainfall events happen immediately after tillage, soil consolidation will proceed much more quickly than the typical average time dependence. In this study, we have linked soil consolidation effects on erosion to soil sealing effects on infiltration to produce a more general way to estimate this crucial factor of soil consolidation.

With a small grid-cell size, PALMS can predict changes in detachment and deposition across short distances as a function of local soil properties and microtopography. As a result of the higher temporal resolution, extreme events can be identified and their importance for total erosion levels can be evaluated. Thanks to the grid approach, PALMS can be used to predict deposition or the pathway taken by the eroded material as it moves in complex landscapes, from hillslope sites to water bodies.

The example simulation from an actual farm field shows the power of PALMS in terms of mapping the erosion, deposition, and sediment yield for complex landscapes. Although various assumptions about rill patterns were tried, for this example the influence of these various patterns was minor.

The above capabilities are contained in a model that also quantifies the spatial and temporal distributions of evapotranspiration, soil evaporation, photosynthesis, plant and soil respiration, infiltration, drainage with and without tiles, crop growth and yield, root water uptake, soil temperature, soil moisture distributions, energy fluxes of radiation and convection, and influences of compaction on soil water properties.

	APPENDIX

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
LIST OF SYMBOLS

ai coefficient used to calculate soil consolidation effects in the interrill, unitless ar coefficient used to calculate soil consolidation effects in the rill, unitless b coefficient used to calculate soil consolidation effects, unitless ci,can interrill erodibility adjustment factor for canopy effects, unitless (Alberts et al., 1995, Eq. 7.10.4) ci,dr interrill erodibility adjustment factor for dead root biomass, unitless (Alberts et al., 1995, Eq. 7.10.6) ci,ft interrill erodibility adjustment factor for freeze–thaw cycles, unitless (Alberts et al., 1995, Eq. 7.10.11) ci,gc interrill erodibility adjustment factor for ground cover, unitless (Alberts et al., 1995, Eq. 7.10.5) ci,sc interrill erodibility adjustment factor for soil consolidation, unitless ci,lr interrill erodibility adjustment factor live root biomass, unitless (Alberts et al., 1995, Eq. 7.10.7) ci,sl interrill erodibility adjustment factor for interrill slope adjustment, unitless (Alberts et al., 1995, Eq. 7.10.8) cr,dr rill erodibility adjustment factor for dead roots, unitless (Alberts et al., 1995, Eq. 7.11.7) cr,ft rill erodibility adjustment factor for freeze and thaw cycles, unitless (Alberts et al., 1995, Eq. 7.11.11) cr,res rill erodibility adjustment factor for incorporated residue, unitless (Alberts et al., 1995, Eq. 7.11.6) cr,lr rill erodibility adjustment factor for live roots, unitless (Alberts et al., 1995, Eq. 7.11.8) cr,sc rill erodibility adjustment factor for soil consolidation, unitless cs,ft critical shear stress adjustment factor for freeze–thaw cycles, unitless (Alberts et al., 1995, Eq. 7.11.16) cs,sc critical shear stress adjustment factor for soil consolidation, unitless (Alberts et al., 1995, Eq. 7.11.14) cs,rr critical shear stress adjustment factor for surface roughness, unitless (Alberts et al., 1995, Eq. 7.11.13) clay clay fraction in soil surface Dc,n detachment capacity by rill flow, kg s–1 m–2 (Foster et al., 1995, Eq. 11.2.3) Di,k,n interrill sediment delivery to the rill for a sediment particle size class k at node n, kg s–1 m–2 (Foster et al., 1995, Eq. 11.3.10) dk particle diameter for a particle size class k, m Dr,k,n soil detachment (+) or deposition (–) in the rill for a sediment particle size class k at node n, kg s–1 m–2 (Foster et al., 1995, Eq. 11.2.2 and 11.2.4) EIaccum accumulated amount of rainfall energy intensity, MJ cm ha–1 h–1 (Wischmeier and Smith, 1978) f total friction coefficient for rill areas, unitless (Gilley and Weltz, 1995) fcr friction coefficient for rill surface residue, unitless (Gilley and Weltz, 1995) flive friction coefficient for living plants that act to slow runoff, unitless (Gilley and Weltz, 1995) Fnozzle adjustment factor to account for sprinkler irrigation nozzle impact energy variation (Foster et al., 1995) fsr friction coefficient for rill surface roughness, unitless g acceleration of gravity, m s–2 Gk,n sediment load for a sediment particle size class k at node n, kg s–1 m–1 Gt,n total sediment load at node n, kg s–1 m–1 Ie effective rainfall intensity, m s–1 (Foster et al., 1995) k individual particle size class of sediment ki interrill erodibility factor, kg s m–4 (Alberts et al., 1995, Eq. 7.10.3) kr rill erodibility factor, s m–1 (Alberts et al., 1995, Eq. 7.11.5) kib baseline interrill erodibility, kg s m–4 (Alberts et al., 1995, Eq. 7.10.1 and 7.10.2) krb baseline rill erodibility, s m–1 (Alberts et al., 1995, Eq. 7.11.1 and 7.11.3) ki,cons consolidated soil erodibility for the interrill, kg s m–4 (Alberts et al., 1995, Eq. 7.10.9) kr,cons consolidated soil erodibility for the rill, s m–1 (Alberts et al., 1995, Eq. 7.11.10) n grid-cell node Qr,n rill flow rate at node n, m3 s–1 Rs rill spacing, m sand sand fraction in soil surface SDRk,n sediment delivery ratio for a sediment particle size class k (Foster et al., 1995) SGk specific gravity for a sediment particle size class k, unitless silt silt fraction in soil surface tadj adjustment factor for the sediment transport capacity for sandy soils, unitless (Foster et al., 1995, Eq. 11.2.9) Tk,n sediment transport capacity for a sediment particle size class k at node n, kg s–1 m–1 Tt,n total sediment transport capacity at node n, kg s–1 m–1 Vf,e effective fall velocity for the sediment, m s–1 Vf,k fall velocity for a sediment particle size class k, m s–1 wn rill width at node n, m x distance downslope, m qn flow discharge per unit width at node n, m2 s–1 ß raindrop-induced turbulence coefficient, dimensionless (Foster et al., 1995) Ek,n+1 net detachment or deposition, kg m–2 t routing time step, s x node spacing, m kinematic viscosity of water, m2 s–1 i,n interrill runoff rate, m s–1 c soil surface critical shear stress, kg m–1 s–2 (Alberts et al., 1995, Eq. 7.11.12) cb soil surface baseline critical shear stress, kg m–1 s–2 (Alberts et al., 1995, Eq. 7.11.2 and 7.11.4) r,n flow shear stress acting on the soil surface particles at node n, kg m–1 s–2

	ACKNOWLEDGMENTS

 
We acknowledge Dr. George Foster for insights on measurement and modeling approaches, Dr. Dennis Flanagan and Dr. Mark Nearing (USDA-NSERL) for assistance with interpretation of WEPP and Earth It, Inc. (Madison, WI) for DEM and soildata in fields. This research was partially supported with funds from the USDA Cooperative State Research, Education, and Extension Service's Water Quality Program under Grant no. 2002-51130-01951, and the Wisconsin Buffer Initiative funded by the Natural Resources Conservation Service under Grants no. 68-5F48-3-085 and 68-5F48-4-082 and the Wisconsin Department of Natural Resources with funding from the Bureau of Watershed Management, Runoff Management Section for Non-Point Source Pollution Abatement.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

Received for publication August 29, 2006.

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TOP NOTES ABSTRACT INTRODUCTION THE PRECISION AGRICULTURAL... RESULTS CONCLUSIONS APPENDIX REFERENCES

 

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